Tutte Polynomial of Ideal Arrangement

TL;DR

This paper extends the computation of Tutte polynomials to ideal arrangements in root systems, improving existing methods and providing explicit formulas for classical and some exceptional root systems.

Contribution

It introduces a refined finite field method for ideal arrangements and computes Tutte polynomials for classical and certain exceptional root systems.

Findings

Computed Tutte polynomials for classical root system ideals.

Improved finite field method applicable to any suitable finite field.

Derived formulas for exceptional root systems G2, F4, and E6.

Abstract

The Tutte polynomial is originally a bivariate polynomial enumerating the colorings of a graph and of its dual graph. But it reveals more of the internal structure of the graph like its number of forests, of spanning subgraphs, and of acyclic orientations. In 2007, Ardila extended the notion of Tutte polynomial to the hyperplane arrangements, and computed the Tutte polynomials of the classical root systems for a certain prime power of the first variable. In this article, we compute Tutte polynomials of ideal arrangements. Those arrangements were introduced in 2006 by Sommers and Tymoczko, and are defined for ideals of root systems. For the ideals of the classical root systems, we bring a slight improvement of the finite field method showing that it can applied on any finite field whose cardinality is not a minor of the matrix associated to a hyperplane arrangement. Computing the minor set associated to an ideal of a classical root system permits us particularly to deduce the Tutte polynomials of the classical root systems. For the ideals of the exceptional root systems of type $G_2$, $F_4$, and $E_6$, we use the formula of Crapo.

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